# The definable subcategory induced by a large canonical module

Auslander and Buchweitz (1989) showed that the class of maximal Cohen-Macaulay modules over a Cohen-Macaulay local ring with a canonical module is part of a complete cotorsion pair in the category of finitely generated modules. As shown by Miyachi (1998), this fact holds more generally for an R-order over a Cohen-Macaulay ring R with a (pointwise) canonical module. On the other hand, Holm (2017) established a perfect cotorsion pair (X, Y) in the category of all modules over a Cohen-Macaulay local ring with a canonical module such that X is the smallest definable subcategory containing all maximal Cohen-Macaulay modules. This result was deduced by showing a Govorov-Lazard type result for X, and the modules in X are those called weak balanced big Cohen-Macaulay. In my talk, I will suggest an infinitely generated version of a canonical module, and explain how this concept makes sense to generalize Holm’s results to a non-commutative and non-local setup like Miyachi’s work. It is also possible to partly avoid the existence of a canonical module, so that some results on balanced big Cohen-Macaulay approximation due to Simon (2009) and Holm (2017) can be unified. This work is inspired by ongoing joint work with Michal Hrbek and Jan Stovicek about large (co)tilting complexes over a commutative noetherian ring, and related to recent joint work with Ryo Kanda about flat cotorsion modules over Noether algebras.